Abstract

A class of linear and nonlinear dynamical problems that arise when studying the modulation of trains of nearly identical soliton pulses of the nonlinear Schrödinger equation is introduced. In the simplest case the dynamics of the nonlinear Schrödinger equation can be reduced to an equation that is a complex extension of the integrable Toda lattice equation, so that the latter asymptotically models the former in the case of large intersoliton separations.

© 1998 Optical Society of America

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References

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  1. J. M. Arnold, “Soliton pulse position modulation,” Proc. Inst. Electr. Eng. 140, 359–366 (1993).
  2. J. M. Arnold, “Digital pulse position modulation of optical fibre solitons,” Opt. Lett. 21, 31–33 (1996).
    [CrossRef]
  3. J. M. Arnold, “Stability of nonlinear pulse trains on optical fibres,” in Proceedings of Electromagnetic Theory Symposium (International Union of Radio Science, Gent, Belgium, 1995), pp. 553–555.
  4. V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
    [CrossRef] [PubMed]
  5. V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
    [CrossRef]
  6. V. Karpman, Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
    [CrossRef]
  7. J. M. Arnold, “Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation,” IMA J. Appl. Math. 52, 123–140 (1994).
    [CrossRef]
  8. A. Hasegawa, Y. Kodama, Optical Fibre Solitons (Cambridge U. Press, Cambridge, UK, 1994).
  9. G. B. Whitham, Linear and Nonlinear Waves (Academic, Orlando, Fla., 1974).
  10. K. A. Gorshkov, L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
    [CrossRef]
  11. D. Anderson, M. Lisak, “Bandwidth limits due to interpulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986).
    [CrossRef]
  12. L. D. Fadeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
  13. J. P. Gordon, H. A. Haus, “Random walk of coherently amplified solitons,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  14. V. E. Zhakarov, A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  15. J. Satsuma, N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
    [CrossRef]
  16. E. J. Hinch, Perturbation Methods (Cambridge U. Press, Cambridge, UK, 1991).
  17. S. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Sov. Phys. JETP 40, 269–274 (1974).
  18. H. Flaschka, D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
    [CrossRef]
  19. H. Flaschka, “The Toda lattice. I. Existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).
    [CrossRef]

1997 (1)

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

1996 (2)

J. M. Arnold, “Digital pulse position modulation of optical fibre solitons,” Opt. Lett. 21, 31–33 (1996).
[CrossRef]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

1994 (1)

J. M. Arnold, “Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation,” IMA J. Appl. Math. 52, 123–140 (1994).
[CrossRef]

1993 (1)

J. M. Arnold, “Soliton pulse position modulation,” Proc. Inst. Electr. Eng. 140, 359–366 (1993).

1986 (2)

1981 (2)

K. A. Gorshkov, L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

V. Karpman, Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

1976 (1)

H. Flaschka, D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
[CrossRef]

1974 (3)

H. Flaschka, “The Toda lattice. I. Existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).
[CrossRef]

J. Satsuma, N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

S. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Sov. Phys. JETP 40, 269–274 (1974).

1972 (1)

V. E. Zhakarov, A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Anderson, D.

Arnold, J. M.

J. M. Arnold, “Digital pulse position modulation of optical fibre solitons,” Opt. Lett. 21, 31–33 (1996).
[CrossRef]

J. M. Arnold, “Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation,” IMA J. Appl. Math. 52, 123–140 (1994).
[CrossRef]

J. M. Arnold, “Soliton pulse position modulation,” Proc. Inst. Electr. Eng. 140, 359–366 (1993).

J. M. Arnold, “Stability of nonlinear pulse trains on optical fibres,” in Proceedings of Electromagnetic Theory Symposium (International Union of Radio Science, Gent, Belgium, 1995), pp. 553–555.

Diankov, G. L.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Evstatiev, E. G.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

Fadeev, L. D.

L. D. Fadeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

Flaschka, H.

H. Flaschka, D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
[CrossRef]

H. Flaschka, “The Toda lattice. I. Existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).
[CrossRef]

Gerdjikov, V. S.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

Gordon, J. P.

Gorshkov, K. A.

K. A. Gorshkov, L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, Optical Fibre Solitons (Cambridge U. Press, Cambridge, UK, 1994).

Haus, H. A.

Hinch, E. J.

E. J. Hinch, Perturbation Methods (Cambridge U. Press, Cambridge, UK, 1991).

Karpman, V.

V. Karpman, Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Kaup, D. J.

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

Kodama, Y.

A. Hasegawa, Y. Kodama, Optical Fibre Solitons (Cambridge U. Press, Cambridge, UK, 1994).

Lisak, M.

Manakov, S.

S. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Sov. Phys. JETP 40, 269–274 (1974).

McLaughlin, D.

H. Flaschka, D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
[CrossRef]

Ostrovski, L. A.

K. A. Gorshkov, L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Satsuma, J.

J. Satsuma, N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zhakarov, A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Solov’ev,

V. Karpman, Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Takhtajan, L. A.

L. D. Fadeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

Uzunov, I. M.

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Academic, Orlando, Fla., 1974).

Yajima, N.

J. Satsuma, N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Zhakarov, V. E.

V. E. Zhakarov, A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

IMA J. Appl. Math. (1)

J. M. Arnold, “Stability theory for periodic pulse train solutions of the nonlinear Schrödinger equation,” IMA J. Appl. Math. 52, 123–140 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (1)

H. Flaschka, “The Toda lattice. I. Existence of integrals,” Phys. Rev. B 9, 1924–1925 (1974).
[CrossRef]

Phys. Rev. E (1)

V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, G. L. Diankov, “Nonlinear Schrödinger equation and N-soliton interactions: generalised Karpman–Solov’ev approach and the complex Toda chain,” Phys. Rev. E 55, 6039–6060 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, E. G. Evstatiev, “Asymptotic behavior of N-soliton trains of the nonlinear Schrödinger equation,” Phys. Rev. Lett. 77, 3943–3946 (1996).
[CrossRef] [PubMed]

Physica D (2)

V. Karpman, Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

K. A. Gorshkov, L. A. Ostrovski, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

J. M. Arnold, “Soliton pulse position modulation,” Proc. Inst. Electr. Eng. 140, 359–366 (1993).

Prog. Theor. Phys. (1)

H. Flaschka, D. McLaughlin, “Canonically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions,” Prog. Theor. Phys. 55, 438–456 (1976).
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma, N. Yajima, “Initial value problem of one- dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284–306 (1974).
[CrossRef]

Sov. Phys. JETP (2)

V. E. Zhakarov, A. B. Shabat, “Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

S. Manakov, “Complete integrability and stochastization of discrete dynamical systems,” Sov. Phys. JETP 40, 269–274 (1974).

Other (5)

L. D. Fadeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

E. J. Hinch, Perturbation Methods (Cambridge U. Press, Cambridge, UK, 1991).

J. M. Arnold, “Stability of nonlinear pulse trains on optical fibres,” in Proceedings of Electromagnetic Theory Symposium (International Union of Radio Science, Gent, Belgium, 1995), pp. 553–555.

A. Hasegawa, Y. Kodama, Optical Fibre Solitons (Cambridge U. Press, Cambridge, UK, 1994).

G. B. Whitham, Linear and Nonlinear Waves (Academic, Orlando, Fla., 1974).

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Equations (81)

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ixψ+½t2ψ+|ψ|2ψ=0
ixΨ+½(t2-1)Ψ+|Ψ|2Ψ=0.
Ψ=sech(t).
S=Ldx,
L=½i-(Ψ*xΨ-ΨxΨ*)dt-H,
H=½-(|tΨ|2+|Ψ|2-|Ψ|4)dt.
Ψ=kZηk exp(iαk)exp[-iωk(t-tk)] ×sech[ηk(t-tk)],
S¯=2k(pkdxtk-ηkdxαk)-H¯dx,
H¯=k(½pk2-½νk2)-4k exp[-(tk+1-tk)]cos(αk+1-αk),
H¯=k½(pk2-νk2)-4k{exp[-(tk+1-tk)] ×cos(αk+1-αk)-exp(-T)cos α},
dxtk=pk,
dxαk=νk,
dxpk=4{exp[-(tk+1-tk)]cos(αk+1-αk)-exp[-(tk-tk-1)]cos(αk-αk-1)},
dxνk=-4{exp[-(tk+1-tk)]sin(αk+1-αk)-exp[-(tk-tk-1)]sin(αk-αk-1)}.
dx2sk=4{exp[-(sk+1-sk)]-exp[-(sk-sk-1)]},
dx2ξk=4 exp(-τ)(2ξk-ξk-1-ξk+1).
ξk=Z exp(iλx)exp(ikβ),
λ2=-8(1-cos β)exp(-τ)=-8(1-cos β)exp(-T)exp(-iα).
Im(λ)=±4 exp(-T/2)sin(β/2)cos(α/2),
L=(1/4)exp(T/2).
ξk(x)=12π-ππ{P(β)sin[λ(β)x]+Q(β)cos[λ(β)x]}exp(ikβ)dβ,
ξk(x)=jZ[ξj(0)J2(k-j)(λ0x)+ξj(0) ×0xJ2(k-j)(λ0x)dx],
Q(β)=jξj(0)exp(ijβ),P(β)=jξj(0)exp(ijβ).
λ0=4i exp(-τ/2)=4i exp(-T/2)exp(-iα/2),
ξk(x, x)=rZ[pr(x)+iνr(x)]0x-xJ2(k-r)(λ0x)dx
ξk(x)=rZ0x[pr(x)+iνr(x)] ×0x-xJ2(k-r)(λ0x)dxdx.
qk2(x)=σp2G+(x)+σν2G-(x),
θk2(x)=σp2G-(x)+σν2G+(x),
G±(x)=rZ0x[|Fr(x)|2±Re Fr2(x)]dx,
Fr(x)=0xJ2r(λ0x)dx,
pr(x)ps(y)=σp2δrsδ(x-y),
νr(x)νs(y)=σν2δrsδ(x-y),
pr(x)νs(y)=0.
qk2(x)σp2x3,
θk2(x)σν2x3.
xM=[A,M],xϕk=Aϕk
M=2it-Ψ*Ψ-it,
A=-i-¼+t2+½|Ψ|2i[Ψ*t+½(tΨ*)]-i[Ψt+½(tΨ)]¼-t2-½|Ψ|2,
Ψk=ηk exp(iαk)exp[-iωk(t-tk)]sech[ηk(t-tk)],
Mk(0)=2it-Ψk*Ψk-it.
Mk(0)ϕk(0)=ζk(0)ϕk(0)
Mˆk(0)ϕˆk(0)=ζk(0)ϕˆk(0),
-ϕ1t(Mϕ2)dt=-ϕ2t(Mˆϕ1)dt
ϕˆk(0)=σ1ϕk(0),σ1=0110.
ϕk(t)=kvkk(t)ϕk(0)(t)
dxM¯=[A¯,M¯].
ϕk(0)(t)=½ ηk1/2 exp[¼ iπ-½ iαk+½ i(ωk+iηk)(t-tk)]sech[ηk(t-tk)]½ ηk1/2 exp[-¼ iπ+½ iαk-½ i(ωk+iηk)(t-tk)]sech[ηk(t-tk)],
ϕk(0)(t)=½ ηk1/2 exp[-¼ iπ+½ iαk-½ i(ωk+iηk)(t-tk)]sech[ηk(t-tk)]½ ηk1/2 exp[¼ iπ-½ iαk+½ i(ωk+iηk)(t-tk)]sech[ηk(t-tk)],
M¯jk=-[ϕˆj(0)tMϕk(0))dt,
B¯jk=-ϕˆj(0)tϕk(0)dt
M¯jk=iB¯jk+R¯jk+o(),
B¯jk=δjk+o(1),
R¯jk=(pk+iνk)δjk+2i exp[-i(αk-αk-1)/2] ×exp[-(tk-tk-1)/2]δj+1k+2i ×exp[-i(αk+1-αk)/2] ×exp[-(tk+1-tk)/2]δj-1k.
ζkB¯vk=[iB¯+R¯+o()]vk,
ζkvk=[i+R¯+o()]vk.
νk=0,αk=kπforallkZ,
A¯jk=-i exp[-i(αk-αk-1)/2] ×exp[-(tk-tk-1)/2]δj+1 k+i ×exp[-i(αk+1-αk)/2] ×exp[-(tk+1-tk)/2]δj-1 k.
dxtk=pk,
dxpk=-4{exp[-(tk+1-tk)]-exp[-(tk-tk-1)]},
Jr=Tr(R¯r)r=1, 2, 3,  N,
{F,G}=k=1NFtkGpk-FpkGtk-k=1NFαkGνk-FνkGαk
{f, g}C=k=1Nfskgσk-fσkgsk
dxsk={sk, h}C,
dxσk={σk, h}C,
Ir=-(i/r)k(ζkr-ζk*r)=2 Im{r-1kζkr},
I1=-|Ψ|2dt,
I2=i-{Ψ*tΨ-ΨtΨ*}dt,
I3=-{|tΨ|2-|Ψ|4}dt,
 etc.
H=(I1+I3)/2.
xM¯=[M¯, A¯],xvk=A¯vk,
xTr(M¯r)=0,
M¯=i+R¯+o(),
M¯=i+R¯.
Tr(M¯r)=irji-jn!/(n-j)!j!Tr(Rj).
J0=N,
J1=kσk,
J2=k{σk2-8 exp[-(sk+1-sk)]}.
H=(I3+I1)/2Im Tr(M¯+M¯3)=-Re Tr[()(1-iR¯)3-(1-iR¯)]=2[()J0+(½)Re(J2)+()Im(J3)],
H=Re(J2)+O(ε3)
H2H¯.

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